We attempted to model static stresses from mounting the cylinder head, thermal stresses, and mechanical stresses from mass forces, combustion forces and the like.
Our goal was to achieve low combustion gas blow-by, low oil ingress into the combustion chamber, yet low friction between piston and cylinder wall.
We used simulations (about three months worth IIRC) and measurements to generate data points for a continuous ersatz model of the physical phenomena, then performed multi-criteria optimisation on that model. The resulting shape was interesting, but not exactly earth-shattering. Kinda randomly warped.
This work was tremendous fun, and I learned a lot.
If anyone is interested (and can read German), it resulted in this dissertation: https://books.google.de/books/about/Tribo_System_Kolbengrupp...
Do you have a couple pictures of the piston handy?
Give me a few hours to dig up my old thesis. That was more than a decade ago!
I'll reply to your post again with an image link once I've found it.
I scanned a picture from the dead-tree version: https://imgur.com/vQLIQ1F
Top image is the cylinder shape without optimisation, bottom is the optimised shape.
Obviously, the contour distortion is massively exaggerated (ISTR a factor of 1000), otherwise it wouldn't have been visible; we're talking micron scale here.
Thanks very much for the follow up!
Thermal mechanical stuff is a big pain. One of my professors mentioned as a young engineer measuring expansion on exhaust manifolds by drawing faint scratches and microscope to measure how far they stretched as the manifold got hot. Important because localized thermally induced stress causes fatigue cracking.
http://blog.jepistons.com/maximizing-horsepower-with-piston-...